Still getting 'too many devices' message

I want to authorise my new Mac to play previous downloads in Music. I had the 'too many devices' message, so removed 2 from my authorised devices list – my account now shows that I only have 3 authorised devices. However, when I try to authorise the new Mac, I keep getting the same message. I've shut down/reopened Music, logged out/back in to my Apple ID and no joy.

Thoughts?